This paper provides new insights into the asymptotic properties of the synthetic control method (SCM). We show that the synthetic control (SC) weight converges to a limiting weight that minimizes the mean squared prediction risk of the treatment-effect estimator when the number of pretreatment periods goes to infinity, and we also quantify the rate of convergence. Observing the link between the SCM and model averaging, we further establish the asymptotic optimality of the SC estimator under imperfect pretreatment fit, in the sense that it achieves the lowest possible squared prediction error among all possible treatment effect estimators that are based on an average of control units, such as matching, inverse probability weighting and difference-in-differences. The asymptotic optimality holds regardless of whether the number of control units is fixed or divergent. Thus, our results provide justifications for the SCM in a wide range of applications. The theoretical results are verified via simulations.
翻译:本文对合成控制方法(SCM)的无症状特性提供了新的洞察力。我们表明,合成控制(SC)重量与限制重量相交,限制重量将治疗效应估计值的平均平方预测风险降到最低,当预处理期达到无限时,我们也将趋同率量化。观察SCM与平均模型之间的联系,我们进一步确定在不完善的预处理条件下SC估计值的无症状最佳性,即它在所有可能的治疗效应估计器中达到尽可能最低的平方预测误差,这些估计器以对照、偏差加权和异差等控制器平均值为基础。无症状最佳性维持着,无论控制器的数目是固定的还是不同的。因此,我们的结果为SCM的广泛应用提供了理由。理论结果通过模拟得到验证。